Binomial coefficient inequality equation proofs
Prove the statement:
If n is a positive integer, then
I tried to read up the falling/rising factorial notation from internet sources, but I cannot find how to write up the first step.
The options are as follows (We have to put it in the right order, one option is incorrect):
- If , then , so the "greater than" signs are correct. Similarly, if , then , so the "less than" signs are correct.
- If , then , so the "less than" signs are correct. Similarly, if , then , so the "greater than" signs are correct.
- Since , the equalities at the ends are clear.
- The equalities at the ends are clear. Using the factorial formulae for computing binomial coefficients, we see that
If we consider a sample k as and we consider .
Then because the floor value of n/2 is 3. But the same holds good on the right side of the equation with greater than signs with because the ceiling value of n/2 is 4.
It is confusing to understand which of the steps is correct between the and provided in the option list. Also what is the first step for solving this, the problem definition or something similar does not exist in the text provided.